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Properties

Label 2-364-28.27-c1-0-27
Degree 22
Conductor 364364
Sign 0.661−0.750i0.661 - 0.750i
Analytic cond. 2.906552.90655
Root an. cond. 1.704861.70486
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 1.32i)2-s + 2.64·3-s + (−1.50 − 1.32i)4-s + (−1.32 + 3.50i)6-s + 2.64·7-s + (2.50 − 1.32i)8-s + 4.00·9-s − 2.64i·11-s + (−3.96 − 3.50i)12-s − i·13-s + (−1.32 + 3.50i)14-s + (0.500 + 3.96i)16-s + (−2.00 + 5.29i)18-s − 5.29·19-s + 7.00·21-s + (3.50 + 1.32i)22-s + ⋯
L(s)  = 1  + (−0.353 + 0.935i)2-s + 1.52·3-s + (−0.750 − 0.661i)4-s + (−0.540 + 1.42i)6-s + 0.999·7-s + (0.883 − 0.467i)8-s + 1.33·9-s − 0.797i·11-s + (−1.14 − 1.01i)12-s − 0.277i·13-s + (−0.353 + 0.935i)14-s + (0.125 + 0.992i)16-s + (−0.471 + 1.24i)18-s − 1.21·19-s + 1.52·21-s + (0.746 + 0.282i)22-s + ⋯

Functional equation

Λ(s)=(364s/2 ΓC(s) L(s)=((0.661−0.750i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(364s/2 ΓC(s+1/2) L(s)=((0.661−0.750i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 364364    =    22⋅7⋅132^{2} \cdot 7 \cdot 13
Sign: 0.661−0.750i0.661 - 0.750i
Analytic conductor: 2.906552.90655
Root analytic conductor: 1.704861.70486
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ364(27,⋅)\chi_{364} (27, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 364, ( :1/2), 0.661−0.750i)(2,\ 364,\ (\ :1/2),\ 0.661 - 0.750i)

Particular Values

L(1)L(1) ≈\approx 1.66967+0.753717i1.66967 + 0.753717i
L(12)L(\frac12) ≈\approx 1.66967+0.753717i1.66967 + 0.753717i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.5−1.32i)T 1 + (0.5 - 1.32i)T
7 1−2.64T 1 - 2.64T
13 1+iT 1 + iT
good3 1−2.64T+3T2 1 - 2.64T + 3T^{2}
5 1−5T2 1 - 5T^{2}
11 1+2.64iT−11T2 1 + 2.64iT - 11T^{2}
17 1−17T2 1 - 17T^{2}
19 1+5.29T+19T2 1 + 5.29T + 19T^{2}
23 1−7.93iT−23T2 1 - 7.93iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+2.64T+31T2 1 + 2.64T + 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1−7iT−41T2 1 - 7iT - 41T^{2}
43 1+5.29iT−43T2 1 + 5.29iT - 43T^{2}
47 1+7.93T+47T2 1 + 7.93T + 47T^{2}
53 1−4T+53T2 1 - 4T + 53T^{2}
59 1+10.5T+59T2 1 + 10.5T + 59T^{2}
61 1−7iT−61T2 1 - 7iT - 61T^{2}
67 1+13.2iT−67T2 1 + 13.2iT - 67T^{2}
71 1−15.8iT−71T2 1 - 15.8iT - 71T^{2}
73 1+7iT−73T2 1 + 7iT - 73T^{2}
79 1+13.2iT−79T2 1 + 13.2iT - 79T^{2}
83 1+5.29T+83T2 1 + 5.29T + 83T^{2}
89 1+14iT−89T2 1 + 14iT - 89T^{2}
97 1+7iT−97T2 1 + 7iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.30805395046880504813244198053, −10.35334785860889719890597355467, −9.224679860579102861636232742460, −8.612819428244674262057812784745, −7.974833284230237124559641151878, −7.20444887064878828040712720508, −5.82283381509617375380046704361, −4.61402973530226840842132696567, −3.39160677536440472396088069310, −1.69852090821316109022626954131, 1.79748004562692399511237543219, 2.61723776692082725441602347467, 4.00834549831180653712377574379, 4.78296993121953392227327505388, 6.99024399111233468039166660603, 8.039738074984828709027939630971, 8.600504301226130164398589540363, 9.315764092225712011077592039947, 10.36170768744278193085685606103, 11.09276228966929639915028359732

Graph of the ZZ-function along the critical line